See also Dirichlet Series ,
Elliptic Function ,
Elliptic Lambda Function ,
Klein's Absolute Invariant ,
Modular
Equation ,
Modular Form ,
Modular
Group Gamma ,
Modular Group Gamma0 ,
Modular Group Lambda
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References Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. Askey, R. "Ramanujan and Hypergeometric and Basic Hypergeometric
Series." In Ramanujan
International Symposium on Analysis: Proceedings of the Ramanujan Birth Centenary
Year International Symposium held in Pune, December 26-28, 1987 Borwein, J. M. and Borwein, P. B.
"Elliptic Modular Functions." §4.3 in Pi
& the AGM: A Study in Analytic Number Theory and Computational Complexity. Rademacher, H. "Zur Theorie
der Modulfunktionen." J. reine angew. Math. 167 , 312-336, 1932. Rankin,
R. A. Modular
Forms and Functions. Schoeneberg,
B. Elliptic
Modular Functions: An Introduction. Weisstein,
E. W. "Books about Modular Functions." http://www.ericweisstein.com/encyclopedias/books/ModularFunctions.html . Referenced
on Wolfram|Alpha Modular Function
Cite this as:
Weisstein, Eric W. "Modular Function."
From MathWorld https://mathworld.wolfram.com/ModularFunction.html
Subject classifications
is meromorphic in the upper
half-plane 
,
for every matrix 
in the modular group Gamma,
has the form
is a modular function, and every modular function can be expressed
as a rational function of 
(Apostol 1997, p. 40). Modular functions are special
cases of modular forms, but not vice versa.
is modular and not identically 0, then the number of zeros
of 
is equal to the number of poles of 
in the closure of the fundamental
region 
(Apostol 1997, p. 34).
